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"... The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in recon ..."

The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N 2) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1383]. In this paper, we observe that one of the standard interpolation or “gridding ” schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).

...cribe an extremely simple and efficient implementation of the nonuniform fast Fourier transform (NUFFT). There are a host of applications of such algorithms, and we refer the reader to the references =-=[2, 6, 8, 11, 13, 14, 17]-=- for examples. We restrict our attention here to one: function (or image) reconstruction from Fourier data as discussed in [6, 8, 11, 14]. Let us begin, however, with a more precise description of the...

"... We study the problem of estimating the best k term Fourier representation for a given frequency-sparse signal (i.e., vector) A of length N ≫ k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomia ..."

We study the problem of estimating the best k term Fourier representation for a given frequency-sparse signal (i.e., vector) A of length N ≫ k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial(k, log N) time. Randomized sublinear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem [24, 25]. In this paper we develop the first known deterministic sublinear time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method [25]. Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in [30]. 1

...ˆ contains no unpredictably energetic and large (relative to the number of desired Fourier coefficients) frequencies then it is more computationally efficient to simply use standard FFT/NUFFT methods =-=[9, 37, 2, 18, 21]-=-. In other applications [36, 34, 38, 39] where sampling costs are of greater concern than reconstruction runtime, even mild oversampling for the sake of faster reconstruction may be unacceptable. In s...

... The computational bottleneck in (12) is calculating the matrix-vector products Ax and A ′ r, where r denotes the residual y − Ax. We previously used the above gradient expression and combined NUFFTs =-=[46]-=- with temporal interpolation based on a “time-segmentation” approximation [5] so as to be able to compute efficiently Ax and A ′ r [21]. We refer to (12) as the “NUFFT approach.” An alternative, mathe...

Fourier-based reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourier-based reprojection methods. We apply a min-max interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.

"... In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a high-resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, ..."

In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a high-resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, requires no new hardware components and allows the aperture to be compressed. It also presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements.

"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."

NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.

"... Abstract—In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is pose ..."

Abstract—In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k-space trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality. Index Terms—Compressed sensing, fast iterative shrinkage/ thresholding algorithm (FISTA), fast weighted iterative shrinkage/ thresholding algorithm (FWISTA), iterative shrinkage/thresholding algorithm (ISTA), magnetic resonance imaging (MRI), non-Cartesian, nonlinear reconstruction, sparsity, thresholded Landweber, total variation, undersampled spiral, wavelets. I.

..., [29]. For these Fourier precomputations, we ; ; ; Fig. 3. Time evolution of the difference in cost function value with respect to the minimizer for several ISTA algorithms. used the NUFFT algorithm =-=[30]-=- that is made available online. For wavelet transforms, we used the code provided online [21]. This Fourier-domain implementation proved to be faster than Matlab’s when considering reconstructed image...

"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."

In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.

...that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid, and then reconstruct by Fourier inversion. In the second body of literature =-=[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]-=- one has data at an equally spaced Cartesian grid, but wishes to evaluate its discrete Fourier transform as a non-equispaced non Cartesian set of frequencies. This problem is known as either the USFFT...

The introduction of compressed sensing methods to speed up image acquisition has received great attention in the Magnetic Resonance Imaging (MRI) community. Compressed sensing exploits the compressibility of medical images to reconstruct unaliased images from undersampled data. Moreover, compressed sensing can be synergistically combined with previously introduced acceleration methods such as parallel imaging, which employs arrays of receiver coils to further increase imaging speed. Over the past three years, we have been working on the combination of compressed sensing and parallel imaging, exploiting the idea of joint multicoil sparsity. In this work, we present a summary of our image acquisition and reconstruction methods for the combination of compressed sensing and parallel imaging, and describe applications to cardiac and body dynamic MRI. Index Terms — Compressed sensing, parallel imaging,

...for k-t SPARSE-SENSE and GRASP. k-t SPARSE-SENSEsreconstruction employs the FFT along the spatialsdimensions as the operator F in Eq. (1) and (2). For GRASP,sF is given by the Non-Uniform FFT (NUFFT) =-=[12]-=- operators(which was previously used to reconstruct non-Cartesian kspace data). We have implemented two algorithms thatssolve Eq. (3): (a) non-linear conjugate gradient and (b) softthresholding.sThe n...

"... Iterative methods for image reconstruction in MRI are useful in several applications, including reconstruction from non-Cartesian k-space samples, compensation for magnetic field inhomogeneities, and imaging with multiple receive coils. Existing iterative MR image reconstruction methods are either u ..."

Iterative methods for image reconstruction in MRI are useful in several applications, including reconstruction from non-Cartesian k-space samples, compensation for magnetic field inhomogeneities, and imaging with multiple receive coils. Existing iterative MR image reconstruction methods are either unregularized, and therefore sensitive to noise, or have used regularization methods that smooth the complex valued image. These existing methods regularize the real and imaginary components of the image equally. In many MRI applications, including T ∗ 2-weighted imaging as used in fMRI BOLD imaging, one expects most of the signal information of interest to be contained in the magnitude of the voxel value, whereas the phase values are expected to vary smoothly spatially. This paper proposes separate regularization of the magnitude and phase components, preserving the spatial resolution of the magnitude component while strongly regularizing the phase component. This leads to a non-convex regularized least-squares cost function. We describe a new iterative algorithm that monotonically decreases this cost function. The resulting images have reduced noise relative to conventional regularization methods. 1.